An extension of Alberti's result to second order derivatives is obtained. Precisely, if $\Omega$ is an open subset of $R^{N}$
and if $f\in L^{1}\left(\Omega;R^{N\times N}\right)$ is symmetric-valued, then there
exist $u\in W^{1,1}\left( \Omega\right)$ with $\nabla u \in BV(\Omega;R^N)$ and a constant $C>0$ depending
only on $N$ such that
\[
D^{2}u=f\,\mathcal{L}^{N}\lfloor\,\Omega+[\nabla u]\otimes\nu_{\nabla
u}\,\mathcal{H}^{N-1}\lfloor\,S(\nabla u),
\]
and
\[
\int_{\Omega}\left u\right +\left \nabla u\right
\,dx+\int_{S\left(
\nabla u\right) \cap\Omega}\left \left[ \nabla u\right] \right
\,d\mathcal{H}^{N-1}\leq C\int_{\Omega}\left f\right \,dx.
\]